The one-dimensional Saint-Venant equations have been considered for the numerical simulation of shallow water flows on complex topography within this report. Although the Saint-Venant equations are relatively simple compared to more sophisticated models such as the Navier-Stokes system, they are nevertheless capable of predicting enough information to be practically used. From a validation perspective, a number of standard test cases exist for the Saint-Venant equations where analytical solutions are known. This report goes on to introduce the shallow water equations and the Saint-Venant system, by showing up physical and mathematical basis behind the model, followed by a deeper look at the conservation laws and simple derivation of shallow water equations from isentropic flows. Then, the Riemann problem for shallow water equations is presented. The concept of finite volume numerical schemes for the Saint-Vanant system is introduced, together with a description of the kinetic approach to build particular numerical schemes. Numerical tests are performed and, finally, a description of the program that has been implemented to solve the Saint-Venant system in view of comparison with experimental data.